摘要
在这篇论文中给出了分别由弹性、压电和热释电三种材料组成的均匀正交各向异性空心圆柱的轴对称平面应变弹性动力学解。对于均匀弹性空心圆柱，首先通过引入一特定函数将非齐次边界条件化为齐次边界条件，然后利用分离变量法将位移减去特定函数的量展开为关于贝塞尔函数和时间函数乘积的级数，并由贝塞尔函数的正交性，导出时间函数的方程，容易求得此方程的解，最终可求得弹性空心圆柱动力学问题的位移解。对于均匀压电和均匀热释电空心圆柱，首先通过引入一特定函数将非齐次力学边界条件化为齐次力学边界条件，然后利用正交展开技术，导出关于时间函数的方程，再结合由初始条件和电学边界条件，将原问题转化为关于一个时间函数(与电位移函数有关)的第二类Volterra积分方程，并运用插值法，首次构造了两个递推公式，可以保证高精度而又快速地得到此积分方程的解，最终可求得原问题的位移、应力、电位移以及电势的响应解；对于层合空心圆柱，运用状态空间法并结合与求解均匀空心圆柱相似的方法，分别对每种材料给出了层合正交各向异性空心圆柱的弹性动力学解；对于功能梯度材料构成的空心圆柱，运用变量替换，将材料常数沿径向按幂函数r~n规律变化的一类特殊功能梯度材料组成的正交各向异性空心圆柱的弹性动力学问题转化为与均匀空心圆柱弹性动力学问题相同的一组方程进行求解，分别对每种材料给出了这种特殊形式功能梯度材料正交各向异性空心圆柱的弹性动力学解。还进一步研究了在轴对称平面应变基础上并考虑轴向应变的弹性动力学问题。对于弹性空心圆柱，通过引入一特定函数将非齐次边界条件化为齐次边界条件，然后利用正交展开技术，导出关于时间函数的方程，再结合初始条件和端部边界条件，将原问题转化为关于一个时间函数(轴向应变)的第二类Volterra积分方程，运用插值法可给出此积分方程的解；对于压电和热释电空心圆柱，利用求解弹性空心圆柱相似的方法，再结合电学边界条件，原问题转化为关于两个时间函数(轴向应变和与电位移有关的函数)的第二类Volterra积分方程组，同样可用插值法来构造相应的递推公式高效地求解此积分方程组。
给出了弹性、压电和热释电三种材料组成的球面各向同性空心球在球对称变形情形的弹性动力学解。对每一种材料组成的空心球，分别研究了均匀、层合以及材料常数沿径向按幂函数r~n规律变化的一类特殊功能梯度材料三种情形。所用的方法与求解空心圆柱问题相似。
给出了均匀各向同性实心圆柱和实心球的弹性动力学解并讨论了实心圆柱和实心球内的动应力集中现象。还给出了压电空心球平衡问题的通解及其应用。
所给出的研究一维动力学问题方法的优点在于：可以避免积分变换，适用于任意厚度的空心圆柱和空心球在径向变形的弹性动力学问题，还可以方便地处理不同边界条件(边界自由、边界固定及边界上作用指定应力等)的问题，而且便于实现数值计算。
The elastodynamic solutions of othotropic hollow cylinder for axisymmetric plane strain problems are obtained in this paper. For homogeneous, othotropic elastic hollow cylinder, firstly, a special function is introduced to transform the inhomogeneous boundary conditions into homogeneous ones. Secondly, by using the method of separation of variables, the quantity that the displacement subtracts the special function is expanded as the multiplication series of Bessel function and the unknown functions of time. Thirdly, by virtue of the orthogonal properties of Bessel functions, the equations about these unknown functions are derived and the solutions are obtained. Finally, the elastodynamic solution of the hollow cylinder is obtained. For homogeneous, piezoelectric and pyroelectric hollow cylinders, by introducing a special function and by using orthogonal expansion technique, the equation about a function with respect to time is derived. Then by means of the initial conditions and electric boundary conditions, the dynamic problems of piezoelectric and pyroelectric hollow cylinders are transferred to a second kind Volterra integral equation about a time function which is related to electric displacement. And by using interpolation method, two recursive formula are constructed, which can be employed to solve the Volterra integral equation of the second kind efficiently and quickly. The solutions of displacement, stresses, electric placement as well as electric potential are finally obtained. For multilayered hollow cylinders, by using state space method and by following the solving procedure for homogeneous hollow cylinder, the elastodynamic solutions of multilayered orthotropic elastic, piezoelectric as well as pyroelectric hollow cylinder are finally obtained. For functionally graded material (FGM) orthotropic hollow cylinder, a special case that the material constants have a power-law dependence on the radial coordinate is considered. By introducing a new dependent variable, the elastodynamic problems of a FGM orthotropic hollow cylinder are then transferred into those of homogeneous ones which can be solved as above mentioned. The solutions of a special functionally graded, elastic, piezoelectric and pyroelectric, orthotropic hollow cylinders are obtained at the end. The elastodynamic problems of the orthotropic hollow cylinders for the case that the axial strain is considered are also investigated. For elastic hollow cylinder, by introducing a special function and by using the orthogonal expansion technique, the equation about a function of time is derived. And by using the initial conditions as well as the end conditions, the dynamic problem is then transferred to a second kind Volterra integral equation about the function of the axial strain with respect to time which can also be solved successfully by the interpolation method. For piezoelectric and pyroelectric hollow cylinders, by following the solving procedure for elastic hollow cylinder and by using the electric boundary conditions, the dynamic problems are transferred to two Volterra integral equations about two functions of time, one is axial strain and the other is related to electric displacement, which can also be solved efficiently and quickly by employing interpolation method.
The elastodynamic solutions of hollow spheres, which are made of elastic, piezoelectric and pyroelectric materials, respectively, for spherically symmetric
problems are also obtained. And for each of the materials, the elsatodynamic solutions of the homogeneous, multilayered as well as a special FGM (the material constants have a power-law dependent on radial coordinate) hollow sphere are presented at the end. The solving method for hollow spheres is just similar with that for hollow cylinders.
The elastodynamic solutions of homogeneous, isotropic solid cylinder and solid sphere are also presented. And the stress-focusing effect phenomena in the solid cylinder and solid sphere are discussed. Additionally, the general solution of piezoelectric hollow sphere for equilibr
引文
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